Deligne–Simpson conjecture for Coxeter connections

Let $G$ be a simple complex group with Lie algebra $\operatorname{\mathfrak{g}}$. Fix a Coxeter $G$-formal type $\mathscr{A}$ of slope $r/h$ with $\gcd(r,h)=1$. A nilpotent orbit threshold is a nilpotent orbit $\mathscr{O}^r\subset\operatorname{\mathfrak{g}}$ such that the Deligne–Simpson problem is governed by the closure order $\succeq$. Deligne–Simpson conjecture for Coxeter connections. There exists a nilpotent orbit $\mathscr{O}^r\subset\operatorname{\mathfrak{g}}$ such that the Deligne–Simpson problem for Coxeter $G$-connections with initial data $\mathscr{A}$ and nilpotent orbit $\mathscr{O}$ has a positive solution if and only if $\mathscr{O}\succeq\mathscr{O}^r$. Moreover, if $r>h$, then $\mathscr{O}^r=0$, so the Deligne–Simpson problem always has a positive solution. This extends the preceding solution for $\operatorname{GL}_n$-connections to simple complex groups; the candidate is stated in the source as a conjectural generalization, while the supplied parser gives no resolution evidence.